Archimedean operator-theoretic Positivstellensätze

We prove a general archimedean positivstellensatz for hermitian operator-valued polynomials and show that it implies the multivariate Fejer–Riesz theorem of Dritschel–Rovnyak and positivstellensätze of Ambrozie–Vasilescu and Scherer–Hol. We also obtain several generalizations of these and related results. The proof of the main result depends on an extension of the abstract archimedean positivstellensatz for ⁎-algebras that is interesting in its own right.